1 Introduction
As the deadline for Brexit approaches, the British newspapers clarify their positions on what they consider to be the best course of action^{1}^{1}1https://bit.ly/2txNgpn. The three possibilities currently under consideration are (1) accepting Theresa May’s deal with the EU (D), (2) leaving the EU with no deal (N), and (3) postponing Brexit or canceling it altogether (R). Among the major newspapers, the most proLeave position is taken by The Daily Telegraph, which is strongly critical of May’s deal, and ranks it below no deal; it is also strongly opposed to any delays, so its ranking can be described as . On the opposite end of the spectrum is The Guardian, which backs remaining in the EU, but views May’s deal as (scarcely) more acceptable than no deal at all, so its ranking is . The Times and The Sun take a more moderate position: both back May’s deal, but The Times views no deal as the most catastrophic option, while The Sun is firmly opposed to any delays to Brexit. Thus, if we order the newspapers according to their political stance, from left to right, the rankings change from to to to .
These preferences have the property that for any two of the available alternatives , if the first newspaper in our order ranks over then all newspapers that rank over appear before all newspapers that rank over , i.e., every pair of alternatives ‘crosses’ at most once. An ordered list of rankings (an election) that has this property is known as singlecrossing (see Section 2 for formal definitions). Singlecrossing elections have many attractive properties and have recently received a lot of attention in computational social choice literature; see, e.g., the survey by ELPtrends ELPtrends. Besides the example in the previous paragraph, there are reallife settings where we expect to observe essentially singlecrossing preferences. For instance, when a country is about to introduce a flat tax rate and the voters are asked to rank several options (say, 25%, 28%, 33%, 45%), they have to consider the tradeoff between the amount they will have to pay and the value of the government services that can be provided at a given level of taxation; if all voters apply this reasoning, the election where the voters are ordered by income should be singlecrossing (this example dates back to mir:j:singlecrossing mir:j:singlecrossing.
However, the examples we described, while singlecrossing in spirit, may not be singlecrossing in a formal sense, because the singlecrossing property is very fragile. For instance, in the Brexit scenario, if we expand the list of newspapers to include a broader sample of publications, we may find leftleaning newspapers that support no deal. In the tax scenario, not all voters may be capable of evaluating the consequences of each choice.
Now, we can easily check whether an election where voters are ordered according to a publicly observable parameter is singlecrossing, simply by looking at all pairs of candidates. However, as argued above, we expect the answer to be ‘no’. A more important—and more challenging!—task is to understand whether this election is close to singlecrossing, i.e., can be made singlecrossing by deleting a few voters or candidates or swapping a few pairs of candidates, as this will tell us whether the observable parameter used to order the voters is relevant for understanding the voters’ preferences. This knowledge is important, e.g., for managing electoral campaigns, as it can be used to identify the segments of voting population that are more likely to support a given candidate (which can help the campaign managers to decide whom to target in a getoutthevote effort).
Our Contribution
We introduce a mapping between elections and graphs that
enables us to use powerful graphtheoretic machinery to analyze
nearly singlecrossing elections. Briefly, given an election
over a set of candidates (i.e., an ordered list of linear orders over ),
we build an undirected graph that has
as its set of vertices, and contains an edge between two candidates and
if and only if and cross more than once in ;
we say that implements . In other words,
the graph documents obstacles that prevent from being singlecrossing;
in particular, an independent set in corresponds to a subset
of candidates such that the restriction of to this subset
is singlecrossing. We then ask which graphs are implementable,
i.e., can be implemented by elections with voters.
We show (Section 5) that we can obtain
any undirected graph in this manner; in fact, the number of voters required is bounded by a linear
function in the size of the graph. However, for any constant there are graphs
that are not implementable. In Section 4
we focus on implementable graphs and obtain
a complete characterization of this class of graphs by relating it to the class
of permutation graphs. We also argue that all implementable
graphs are comparability graphs;
importantly, every comparability graph is a perfect graph.
Our results have implications for the problem of deciding whether an election is nearly singlecrossing with respect to the given order of voters (Section 6). In particular, we use our mapping to establish the hardness of computing two measures of how close a given election is to being singlecrossing: one of these measures is based on deleting as few candidates as possible, and the other is based on splitting the candidates into as few groups as possible. On the other hand, our results for implementable graphs enable us to show that the problems we consider are in P for elections with voters.
Related Work
The concept of singlecrossing elections has been proposed
in the social choice literature several decades
ago [Mirrlees1971, Roberts1977]. Singlecrossing elections
are appealing both from a purely social choicetheoretic perspective and from
a computational perspective: for instance, their weak majority relation
is necessarily transitive [Mirrlees1971]
and they admit efficient algorithms for determining
a winning committee under a wellknown committee selection rule whose output
is hard to compute for general preferences [Skowron et al.2015].
Several groups of authors have considered the problem of identifying
nearly singlecrossing elections [Bredereck et al.2016, Cornaz et al.2013, Elkind and Lackner2014, Jaeckle et al.2018],
but in all these papers the authors assumed that there was no publicly observable
parameter that determined the ordering of the voters, i.e., they
considered the problem of reordering the voters so that the resulting election
can be made singlecrossing by applying a small number of modifications;
in contrast, we assume that the order of voters is fixed.
Our analysis is similar in spirit to the research on implementation of directed graphs as majority graphs. In this line of work, the input is a directed graph with a vertex set , and the goal is to construct an election over the set of candidates such that there is a directed edge from to in the input graph if and only if a strict majority of voters in prefer to . The classic McGarvey theorem [McGarvey1953] establishes that every directed graph can be implemented in this way using at most two voters per edge, and subsequent work has reduced this number to [Stearns1959, Erdos and Moser1964]; Corollary 9 in Section 5 can be viewed as an analogue of McGarvey’s theorem in our setting. Recently, brandtfew brandtfew investigated what directed graphs can be implemented by elections with two or three voters; this research is similar to our analysis in Section 4.
2 Preliminaries
For each , we denote the set by .
Elections An election is a pair , where is a finite set of candidates and is a list of votes. Each vote , , is a linear order over . We refer to elements of as voters; thus, is the vote of voter . We will sometimes use the term ‘profile’ to refer to , and we use the terms ‘vote’, ‘preference’ and ‘ranking’ interchangeably. We say that voter prefers to or ranks over (and write ) if precedes in the linear order . A restriction of an election with to a subset of candidates is an election , where and for every pair of candidates and every it holds that is ranked above in if and only if is ranked above in .
Singlecrossing (also known as intermediate or orderrestricted) preferences capture settings where the voters can be ordered along a single axis according to their preferences.
Definition 1.
An election with is singlecrossing if for every pair of candidates with there is a such that .
We emphasize that we define singlecrossing elections with respect to a fixed order of the voters, i.e., we are interested in settings where voters are ordered according to a publicly observable parameter.
Graphs An undirected graph is a pair , where is a finite set, and is a collection of size subsets of . The elements of are called vertices, and the elements of are called edges. For readability, we will sometimes write instead of . We assume that the reader is familiar with the definitions of a path, a cycle, a tree, and a bipartite graph. A hole is a cycle with such that if and only if or . An antihole is a sequence of distinct vertices with such that if and only if or .
A directed graph is a pair , where is a finite set, and
is a collection of ordered pairs of elements of
. An element of is called an arc; an arc points from vertex to vertex . An undirected graph can be turned into a directed graph by choosing an orientation for each edge , i.e., transforming into or . A directed graph is said to be transitive if for every triple of vertices such that and it holds that .3 SingleCrossing Implementation
A pair of candidates is a multicrossing pair in an election with if , , and for some with .
Definition 2.
The multicrossing graph of an election is an undirected graph such that and if and only if is a multicrossing pair in . An election implements an undirected graph if . We say that a graph is implementable if there exists an voter election that implements it. Since the set of candidates in an election that implements a graph is necessarily , we often omit from the notation and speak of a profile that implements .
By definition, the only graph that is implementable is the graph with no edges. Thus, in the remainder of the paper we study graphs that are implementable for .
4 Implementability
To build the reader’s intuition, we first consider implementation. We show how to implement several families of graphs, such as paths, trees and evenlength cycles. While for some of these families their implementability follows from the more general results in Section 4.2, the proofs below provide efficient algorithms for finding a voter profile that implements a given graph. Also, we relate implementable graphs to other wellknown classes of graphs, such as permutation graphs and comparability graphs. Finally, we prove that some graphs are not implementable.
4.1 Examples
First, it is easy to see that we can implement empty graphs and cliques: an empty graph is implemented by a profile where all three voters are identical, and a clique can be implemented by a profile where the first and the third voter rank the candidates in the same order, and the second voter ranks the candidates in the opposite order. A somewhat more complex construction establishes that all paths and evenlength cycles are implementable.
Proposition 1.
There is a polynomialtime algorithm that given a graph that is a path or an evenlength cycle constructs a voter profile that implements .
Proof.
Suppose that is a path. For convenience, we assume that and . To start, we construct a voter profile where all voters rank the candidates as . Then, we modify the preferences of the first and the third voter by swapping candidates and in her rankings, for . Also, we modify the preferences of the second voter by swapping candidates and in her ranking, for . Table 1 (left) illustrates the resulting profiles for . To see why this profile implements a path, consider an evennumbered candidate , . By construction, all voters rank above all candidates with and below all candidates with . On the other hand, voters 1 and 3 rank below and above , whereas voter 2 ranks above and below . Thus, is a multicrossing pair if and only if
. For oddnumbered candidates
with , as well as for candidates and , the argument is similar.1  2  1 
3  1  3 
2  4  2 
5  3  5 
4  6  4 
6  5  6 
1  2  3 
3  4  5 
2  3  4 
5  6  1 
4  5  2 
6  1  6 
A similar approach can be used if is an evenlength cycle; we omit the proof, but provide an example in Table 1(right). ∎
The reader may wonder why we only consider cycles of even length in Proposition 1. Now, the cycle of length is implementable because it is a clique. However, for the cycle of length is not implementable; this follows from Theorem 5 in Section 4.2.
On the other hand, we can extend the result of Proposition 1 to arbitrary trees.
Proposition 2.
There is a polynomialtime algorithm that given a graph that is a tree constructs a voter profile that implements .
Proof.
Given a tree , we pick an arbitrary vertex to be its root. Our implementation is recursive. We observe that an isolated vertex is trivially implementable. Then we consider a vertex of whose children are , . We show that, if for each we have a implementation of the tree rooted at in which the first voter ranks first, then we can construct a implementation of the tree rooted at in which the first voter ranks first. Using this idea, we can construct an implementation of starting from the leaves and ending at .
Fix a vertex that is not a leaf. Let be the children of , and for each let be the subtree of rooted at ; let be the set of vertices of . Suppose that for each we have a implementation of in which is ranked first in the first vote.
We will first stack these three implementations on top of each other: we construct a voter profile where each voter ranks all candidates in above all candidates in for all , and for each and each the th voter ranks above if and only if the th voter in the given implementation of ranks above . Then we pull the candidates to the top of the first vote: we modify the preferences of voter so that she ranks in position for and the relative order of the other candidates remains unchanged. Note that this step does not introduce any multicrossing pairs.
In remains to insert into the voters’ rankings. To this end, in the first vote we insert after the first candidates, in the second vote we place on top, and in the third vote we place last. Note that for each the pair is multicrossing, but for every and every the pair is not multicrossing, as both of the first two voters rank above . Thus, we have implemented the edges connecting to its children. However, in the resulting profile voter does not rank first. To remedy this, we first reverse the order of candidates in each vote and then reverse the order of votes; neither of these operations changes the set of multicrossing pairs, and in the resulting profile is ranked first. Each of these steps can be implemented in polynomial time. ∎
4.2 General Constructions
We can relate implementable graphs to two wellknown classes of graphs: permutation graphs and comparability graphs.
Definition 3 (Permutation graph).
An undirected graph is a permutation graph if there exist permutations of such that if and only if appears before in exactly one of the permutations and .
It can be decided in polynomial time whether a given graph is a permutation graph; moreover, if the answer is ‘yes’, the respective permutations can be constructed in polynomial time as well [Golumbic1980, Simon and Trunz1994].
It is immediate that every permutation graph can be implemented by a voter profile.
Theorem 3.
Every permutation graph is implementable, and a voter profile that implements it can be computed in polynomial time.
Proof.
Let be a permutation graph, and let and be two permutations that witness this. Then the profile implements . ∎
In fact, the proof of Theorem 3 suggests a stronger claim: an undirected graph is a permutation graph if and only if it can be implemented by a voter profile where the first and the third voter have the same preferences. Recall that our implementation of cliques has this property, but our implementation of evenlength cycles does not. There is a reason for this: it is not hard to show that cycles of length at least are not permutation graphs. Thus, permutation graphs form a proper subclass of implementable graphs. The following proposition further clarifies the relationship between implementable graphs and permutation graphs.
Proposition 4.
A graph is implementable if and only if there exist two permutation graphs and such that .
Proof.
Let and be two permutation graphs on the same set of vertices . Let (respectively, ) be a pair of permutations witnessing that (respectively, ) is a permutation graph. Note that if a pair of permutations witnesses that a given graph is a permutation graph, then so does the pair of permutations , for any given permutation . Applying this observation to with , we can assume that . Then the profile where for each voter ranks the candidates according to implements : a pair of candidates is multicrossing in this profile if and only if both and disagree with on the order of and .
Conversely, let be a implementation of a graph . Consider the graphs and implemented, respectively, by and . As argued in the proof of Theorem 3, both of these graphs are permutation graphs, and a pair of candidates is multicrossing in if and only if and . This completes the proof. ∎
Another relevant class of graphs is comparability graphs.
Definition 4 (Comparability graph).
A graph is a comparability graph if edges in can be oriented so that the resulting directed graph is transitive.
Comparability graphs can be recognized in polynomial time, and the respective edge orientation can be computed efficiently [Golumbic1980, Simon and Trunz1994].
Theorem 5.
Every implementable graph is a comparability graph.
Proof.
Consider a graph implemented by a profile . We orient the edge from to if and from to otherwise. Since is a multicrossing pair, implies . Consider a pair of arcs , in the resulting directed graph. We have , and hence . Similarly, , implies and , implies . Thus, our directed graph also contains the arc . ∎
Comparability graphs are known to be perfect graphs [Mirsky1971], i.e., graphs that contain neither oddlength holes nor oddlength antiholes^{2}^{2}2Originally, perfect graphs are defined as graphs with the property that the chromatic number of every induced subgraph is equal to the size of the maximum clique in that subgraph [Berge1961]; however, by the strong Berge conjecture, which was proved by perfect perfect, perfect graphs are exactly the graphs with no oddlength holes and no oddlength antiholes.. Hence, Theorem 5 explains why Proposition 1 does not extend to odd cycles: by definition, odd cycles are not perfect graphs. Also, it subsumes the existence results of Propositions 1 and 2: paths, evenlength cycles, and trees can be easily seen to be comparability graphs (we note, however, that these propositions also provide efficient algorithms to compute the respective voter profiles, and it is not clear how to extract such algorithms from the proof of Theorem 5). In particular, every bipartite graph is a comparability graph (we can direct the edges from one part to the other), and paths, evenlength cycles and trees are bipartite graphs. However, there exists a bipartite graph that is not implementable (and hence implementable graphs form a proper subclass of comparability graphs).
Proposition 6.
The bipartite regular graph with parts of size each (see Figure 1) is not implementable.
5 Implementation for
We have seen that not all graphs are implementable. However, we will now show that every graph is implementable by an election whose number of voters is linear in . We first define a class of singlecrossing elections that can be used to implement an arbitrary graph.
Definition 5.
A singlecrossing election with is fully singlecrossing if for every pair of candidates with there is an such that , , and voter ranks just above .
Note that in a fully singlecrossing election the ranking of the last voter is the inverse of the ranking of the first voter, i.e., every pair of candidates ‘crosses’ exactly once.
Theorem 7.
If there exists a fully singlecrossing election with , then every vertex graph is implementable.
Proof.
Consider a fully singlecrossing election with , , and let be an vertex graph. Let . By construction, the election is singlecrossing. Now, for each edge we identify an such that in we have , , and voter ranks just above . We then swap and in the preferences of the st voter in (who, like voter in the original election, ranks just above prior to the swap). This ensures that is a multicrossing pair in the resulting election. In the end we obtain an election that implements . ∎
A fully singlecrossing election with candidates and voters can be obtained as a maximal chain in a weak Bruhat order; in this election, which we will denote by , each vote differs from its predecessor by exactly one swap of adjacent candidates (see [Bredereck et al.2013] and the references therein). One can use as a starting point to implement an arbitrary graph with votes: we take , remove each vote that is obtained from its predecessor by swapping a pair of candidates that does not correspond to an edge in the input graph, and then use the construction in the proof of Theorem 7. However, for dense graphs this produces an implementation with voters.
In contrast, our next theorem, in conjunction with Theorem 7, shows that every graph is implementable. Our construction is inspired by the concept of oddeven sort [Lakshmivarahan et al.1984] and, to the best of our knowledge, is new.
Theorem 8.
For every there exists a fully singlecrossing election with candidates and voters.
Proof.
Let . Consider the following sequence of votes. The first vote is given by . Then, for each , the vote is obtained from the vote by swapping the candidates in positions and for . Similarly, for each , the vote is obtained from the vote by swapping candidates in positions and for . The resulting profile for is given in Table 2. We will now argue that this procedure produces a fully singlecrossing profile.
1  2  2  4  4  6  6  7 
2  1  4  2  6  4  7  6 
3  4  1  6  2  7  4  5 
4  3  6  1  7  2  5  4 
5  6  3  7  1  5  2  3 
6  5  7  3  5  1  3  2 
7  7  5  5  3  3  1  1 
First, we show that in the candidates are ranked as . Indeed, suppose that is even. Then, by construction, for , in the th vote is ranked in position . Then, in vote candidate remains ranked in position , and in the remaining votes moves down step by step, ending up in position . Conversely, if is odd, it moves down step by step in the first votes, stays in the last position for one more step, and then starts climbing back up, ending in position . This implies our claim.
We have shown that each pair of candidates is swapped at least once. To see that it is swapped exactly once, we compute the total number of swaps. If is even, then every evennumbered vote differs from its predecessor by swaps and every oddnumbered vote apart from the first vote differs from its predecessor by swaps. Thus, the total number of swaps is , and hence each pair of candidates is swapped exactly once. For odd values of the calculation is similar.
It remains to note that, by construction, if , but , then is ranked just above in , as we only swap adjacent candidates. This completes the proof. ∎
Corollary 9.
An undirected graph is implementable.
The bound in Corollary 9 is linear in . One can ask if we implement each graph using a constant number of votes. It turns out that the answer is ‘no’.
To show this, we use the Erdös–Szekeres theorem [Erdös and Szekeres1935] to argue that if a graph is implementable by an election with a few voters then it has to have a large clique or a large independent set.
Lemma 10.
If an vertex graph is implementable then it has a clique or size at least or an independent set of size at least .
On the other hand, we have the following wellknown fact, which can be easily proved by the probabilistic method (see, e.g., [Bollobás and Erdös1976]).
Lemma 11.
There exists an integer constant such that for every positive integer there exists a graph with vertices with the property that each clique and each independent set in have at most vertices.
Together, Lemmas 10 and 11 imply that for every there are graphs that are not implementable; in fact, our proof shows that for each there is a graph of size at most with this property.
Theorem 12.
For every positive integer there exists a graph with that is not implementable.
6 Applications
We will now apply the tools developed in Sections 4 and 5 to the problem of detecting elections that are close to being singlecrossing with respect to a given order of voters, for two measures of closeness.
Definition 6.
An instance of Candidate Deletion is given by an election and an integer . It is a yesinstance if and only if there is a subset with such that is singlecrossing. An instance of Candidate Partition is given by an election . It is a yesinstance if and only if can be partitioned into sets so that for each the election is singlecrossing.
We will now show that both of these problems are hard, by leveraging our observation that is singlecrossing if and only if is an independent set in .
Theorem 13.
Candidate Deletion is NPcomplete; Candidate Partition is NPcomplete for every .
Proof.
It is immediate that both of these problems are in NP. To show that Candidate Deletion is NPhard, we reduce from Independent Set. An instance of Independent Set is given by a graph and an integer ; it is a yesinstance if has an independent set of size at least and a noinstance otherwise. This problem is wellknown to be NPhard [Garey and Johnson1979]. Given an instance of the Independent Set problem, we build an election that implements it using the construction described in Section 5; the size of the resulting election is polynomial in the size of , and has an independent set of size at least if and only if is a yesinstance of Candidate Deletion.
We use the same argument for Candidate Partition; the only difference is that we reduce from the Coloring problem. An instance of Coloring is given by a graph ; it is a yesinstance if there exists a mapping such that for every and a noinstance otherwise. Note that each ‘color’ , , forms an independent set in . The Coloring problem is wellknown to be NPhard for every [Garey and Johnson1979]. Again, given a graph , we construct an election that implements it, and observe that is colorable if and only if is a yesinstance of Candidate Partition. ∎
On the other hand, we can use the results in Section 4 to show that Candidate Deletion and Candidate Partition are in P for elections with at most voters.
Theorem 14.
Given an election with at most three voters and an integer , we can decide in polynomial time whether the pair is a yesinstance of Candidate Deletion. Also, for each we can decide in polynomial time whether is a yesinstance of Candidate Partition.
Proof.
Given an election with at most three voters, we construct its multicrossing graph . By Theorem 5 the graph is a comparability graph and hence a perfect graph. As argued in the proof of Theorem 13, to decide whether is a yesinstance of Candidate Deletion, it suffices to determine whether is a yesinstance of Independent Set, and to decide whether is a yesinstance of Candidate Partition, it suffices to determine whether is a yesinstance of Coloring. It remains to note that both Independent Set and Coloring are known to be polynomialtime solvable on perfect graphs (see, e.g., diestel diestel). ∎
By a similar argument, Candidate Partition is polynomialtime solvable for any number of voters.
Proposition 15.
Candidate Partition is polynomialtime solvable.
Proof.
An election is a yesinstance of Candidate Partition if and only if the graph is colorable, and colorability can be checked in polynomial time. ∎
7 Conclusions
We have introduced the notion of singlecrossing implementation of a graph and showed how to exploit the connection between elections and graphs to better understand the complexity of detecting elections that are nearly singlecrossing with respect to a fixed order of voters. Our approach turned out to be useful for two distance measures: the number of candidates that need to be deleted to make the input election singlecrossing, and the number of parts that the candidate set needs to be split into so that the projection of the input election onto each set is singlecrossing. There are other distance measures that can be used in this context: e.g., we can remove or partition voters, or swap adjacent candidates in voters’ preferences. In a companion paper [Lakhani et al.2019], we explore the complexity of computing how far a given election is from being singlecrossing according to several other distance measures.
Our work suggests several interesting open questions. First, it is not known what is the smallest value of such that every vertex graph is implementable: there is a significant gap between the upper bound of Corollary 9 and the lower bound of Theorem 12. Second, our characterization of implementable graphs does not suggest an efficient algorithm for checking whether a graph is implementable. More broadly, we do not know if one can efficiently compute the smallest profile that implements a given graph; we conjecture that this problem is NPcomplete.
References
 [Bachmeier et al.2017] G. Bachmeier, F. Brandt, C. Geist, P. Harrenstein, K. Kardel, D. Peters, and H. G. Seedig. majority digraphs and the hardness of voting with a constant number of voters. Technical report, arXiv 1704.06304, 2017.
 [Berge1961] C. Berge. Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wissenschaftliche Zeitschrift, page 114, 1961.
 [Bollobás and Erdös1976] B. Bollobás and P. Erdös. Cliques in random graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 80(3):419–427, 1976.
 [Bredereck et al.2013] R. Bredereck, J. Chen, and G. Woeginger. A characterization of the singlecrossing domain. Social Choice and Welfare, 41(4):989–998, 2013.
 [Bredereck et al.2016] R. Bredereck, J. Chen, and G. J. Woeginger. Are there any nicely structured preference profiles nearby? Mathematical Social Sciences, 79:61–73, 2016.
 [Chudnovsky et al.2006] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The strong perfect graph theorem. Annals of Mathematics, 164:51–229, 2006.

[Cornaz et al.2013]
D. Cornaz, L. Galand, and O. Spanjaard.
Kemeny elections with bounded singlepeaked or singlecrossing width.
In
Proceedings of the 23rd International Joint Conference on Artificial Intelligence
, pages 76–82, 2013.  [Diestel2012] R. Diestel. Graph Theory. Springer, 2012.
 [Elkind and Lackner2014] E. Elkind and M. Lackner. On detecting nearly structured preference profiles. In Proceedings of the 28th AAAI Conference on Artificial Intelligence, pages 661–667, 2014.
 [Elkind et al.2017] E. Elkind, M. Lackner, and D. Peters. Structured preferences. In U. Endriss, editor, Trends in Computational Social Choice, chapter 10, pages 187–207. AI Access, 2017.
 [Erdos and Moser1964] P. Erdos and L. Moser. On the representation of directed graphs as unions of orderings. Publications of the Mathematical Institute of the Hungarian Academy of Science, 9:125–132, 1964.
 [Erdös and Szekeres1935] P. Erdös and G. Szekeres. A combinatorial problem in geometry. Compositio mathematica, 2:463–470, 1935.
 [Garey and Johnson1979] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NPCompleteness. W. H. Freeman, 1979.
 [Golumbic1980] M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press [Harcourt Brace Jovanovich, Publishers], New YorkLondonToronto, Ont., 1980. With a foreword by Claude Berge, Computer Science and Applied Mathematics.
 [Jaeckle et al.2018] F. Jaeckle, D. Peters, and E. Elkind. On recognising nearly singlecrossing preferences. In Proceedings of the 32nd AAAI Conference on Artificial Intelligence, pages 1079–1086, February 2018.
 [Lakhani et al.2019] F. Lakhani, D. Peters, and E.Elkind. Correlating preferences and attributes: nearly singlecrossing profiles. In Proceedings of the 28th International Joint Conference on Artificial Intelligence, 2019.
 [Lakshmivarahan et al.1984] S. Lakshmivarahan, S. K. Dhall, and L. L. Miller. Parallel sorting algorithms. In Advances in Computers, volume 23, pages 295–354. Elsevier, 1984.
 [McGarvey1953] D. C. McGarvey. A theorem on the construction of voting paradoxes. Econometrica, 21(4):608–610, 1953.
 [Mirrlees1971] J. Mirrlees. An exploration in the theory of optimal income taxation. Review of Economic Studies, 38:175–208, 1971.
 [Mirsky1971] L. Mirsky. A dual of Dilworth’s decomposition theorem. The American Mathematical Monthly, 78(8):876–877, 1971.
 [Roberts1977] K. W. S. Roberts. Voting over income tax schedules. Journal of Public Economics, 8(3):329–340, 1977.
 [Simon and Trunz1994] K. Simon and P. Trunz. A cleanup on transitive orientation. In Orders, algorithms, and applications (Lyon, 1994), volume 831 of Lecture Notes in Comput. Sci., pages 59–85. Springer, Berlin, 1994.
 [Skowron et al.2015] P. Skowron, L. Yu, P. Faliszewski, and E. Elkind. The complexity of fully proportional representation for singlecrossing electorates. Theoretical Computer Science, 569:43–57, 2015.
 [Stearns1959] R. Stearns. The voting problem. The American Mathematical Monthly, 66(9):761–763, 1959.